I understand that the modern notion of Boolean algebra was adapted by Peirce, Schroder and others from Boole's logic of classes. Boole spoke of 1 and 0 as representing the Universe of Discourse and Nothing respectively, while modern Boolean algebra use 1 and 0 as truth-values.
What is the relationship between Universe and the positive truth-value, and Nothing and the negative truth-value? How did Boole's concepts become adapted into truth-values?
As a bonus question, how similar are Boole's logic and Boolean algebra respectively to the first set theories (such as the naive set theories of Frege and Cantor)?
As a point of note, my education is in philosophy rather than mathematics and am undertaking research on the foundations of mathematics, so my mathematical knowledge is relatively limited, though not non-existent. Please bear this in mind when providing you answer. Kind regards!
My guess:
Something being true is equivalent to saying that some item x is a member of some set P or is not a member of some set P, and this is equivalent to saying that item x is a member of the universal set 1, as follows from the law of excluded middle. Furthermore, something being false is equivalent to saying that some item x is a member of some set P and not a member of some set P, which is equivalent to saying that x is a member of the empty set 0, as follows from the law of non-contradiction. Thus, to say that something is true is to say that it fulfills the law of excluded middle, and to say that something is false is to say that it violates the law of non-contradiction.
True ≡ x ∈ P ∨ x ∉ P ≡ x ∈ 1 False ≡ x ∈ P ∧ x ∉ P ≡ x ∈ 0
